Using kernel methods to visualise crime data

Dr. Kieran Martin and Dr. Martin Ralphs

Office for National Statistics, UK

Corresponding author: Kieran Martin, email: kieran.martin@ons.gov.uk,

Abstract

Kernel smoothing methods can be used to better visualise data that are available over geography. The visualisations created by these methods allow users to observe trends in the data which may not be apparent from numerical summaries or point data.

Crime data in the UK has recently been made available to the public at post code level, with visualisations of the amount of recorded events at each postcode for a given area available online. In this paper we apply existing kernel smoothing methods to this data, to assess the use of kernel smoothing methods as applied to open access data.

Kernel smoothing methods are applied to crime data from the greater London metropolitan area, using methods freely available in R.

The kernel smoothers used are demonstrated to provide effective and useful visualisations of the data, and it is proposed that these methods should be more widely applied in data visualisations in the future.

Keywords: smoothing; open data; data visualisation; bandwidth;

1. Introduction

With the increasing level of access to data, users have an unprecedented access to statistics produced across government. There are currently over 8000 data sets available at data.gov.uk. This increased access presents issues with confidentiality; releasing such data should not compromise the identity for individuals or businesses from which statistics are gathered. This issue has been discussed in a government white paper, Unleashing the Potential (2012). The government wishes to provide more transparent data, but avoid compromising people’s identity. Large data releases are also likely to contain mistakes (see Hand, 2012). Approaches to releasing administrative data thus far tend to have focused on releasing data assigned to a specific location, released in a less disclosive form by attaching multiple records to one geographical location.

Raw point estimates can be difficult to understand, especially when they are given over time or space. Simple data summaries (means over an area) can improve comprehension, but doing so can remove some of the spatial patterns that were of interest initially, as well as privilege particular geographical boundaries. One method of summarising data over geography which avoids this problem is to “smooth” the data, creating summaries over small geographical areas which blend together naturally. There have been some investigations into this by the Office for National Statistics (ONS); Carrington et al (2007), produced smoothed estimates for mortality where estimates were smoothed by simple averages, and more recently Ralphs (2011) applied kernel smoothing methods to business data.

Kernel density estimation is a so called “non-parametric” method for summarising data gathered over one or more dimensions. It can be used in multiple contexts, such as estimating the shape of a density function, or looking for underlying trends in data. Kernel density estimation is a descriptive measure, applying to observed data, and does not necessarily provide accurate predictions for future data as parametric methods can.

One use of kernel density estimation can be to visualise patterns in spatial data, such as levels of unemployment or poverty across the country. In this paper we illustrate the use of kernel smoothing methods by application to a publically available data set. Smoothed data will naturally reduce how easy it is to identify individuals, and help control against mistakes in releases by averaging out errors.

The British police keep a record of all reported crime in each constabulary. This is one method used in the United Kingdom for recording crime levels; the British Crime Survey is also used to supplement recorded crime by surveying people’s experience of crime in the last 12 months.

For the last two years police.uk has released monthly information on the number of criminal events that have been recorded in police authorities in England and Wales. This release combines all crimes recorded across that month, and the events are then associated with the centre of the postcode in which said crime was reported to have occurred. While this is a useful visualisation at a local level, as of yet there is no real summary of crime statistics across an area, meaning like with like comparisons are difficult. Comparing one month with another in a particular area can only be done by examining the raw totals of crimes recorded for an area.

This paper describes the application of kernel smoothing methods to crime data. In section 2 the general theory of the kernel density methods used are briefly described. Section 3 presents results for crime across the greater London metropolitan area as well as for a smaller area, in Camden. Section 4 details conclusions and proposals for further investigation.

2. Kernel density estimation for spatial point process patterns

Kernel density estimation involves applying a function (known as a “kernel”) to each data point, which averages the location of that point with respect to the location of other data points. The method has been extended to multiple dimensions and provides an effective method for estimating geographical density. There is a good deal of literature discussing kernel density estimation, such as Simonoff (1996); kernel density estimation was applied in this paper using the package spatstat in R (for more details, see Baddeley and Turner, 2005).

In kernel density estimation, a smooth, curved surface in the form of a kernel function is placed at each data point. The value of the kernel function is highest at the location of the point, and diminishes with increasing distance from the point. Now follows a brief introduction to the mathematical underpinnings of kernel estimation. Given a data set (x1,...,xn)T where xi=(xi1,...,xid)T, d being the dimension being averaged over. Kernel density estimation uses a pre-defined kernel, K, and a

bandwidth matrix H to produce an estimate of the density of said data. The estimate of the density of the data at a point x is then:

n

ii=1

1F ( )= ( - ),n ∑ Hx K x x

where,

1/ 2 1/ 2( ) | | ( ).− −=HK x H K H x

This is effectively a weighted average over an area determined by the bandwidth matrix, H, where observations closer to the centre point are weighted more highly. How much individual observations are weighted depends on the choice of the kernel. The Gaussian kernel was chosen for this investigation as it is commonly used and is the kernel used in spatstat. It has the following form:

/ 2 T1( ) (2 ) exp( ).2

dπ −= −K x x x

This kernel weights observations by their Euclidian distance from x, with the weight reducing at an exponential rate as the distance increases.

The data considered in this paper can be considered to be a spatial point process. This is a data set distributed in 2 dimensions (or three, with time), that has an additional “intensity” attribute at each observation. For crime data, this intensity is the number of incidents observed at that point in space: other examples might be the recorded heights of trees at particular locations.

The intensity of the crime can be smoothed over space using kernel methods, by calculating weighted kernel estimates via the following formula:

( )

( )

i ii

ii

m −

−

∑∑

H

H

K x x

K x x,

where mi is the intensity associated with xi. This is known as the Nadaraya-Watson estimator (Nadaraya, 1964).

Bandwidth selection can be performed manually by examining the output and selecting an appropriate value, or by using various automated methods (see Sheather, 1992). Investigation into possible automated methods, such as Diggle’s method (see Diggle, 1985) found that the bandwidths obtained over smoothed the results at small geographies, and were computationally difficult to obtain at high geographies. It was found that Scott’s rule of thumb (Scott 1992, page 152) provided good values for the bandwidth and was easy to quickly calculate. Scott’s rule has the form:

1/( 4) ,dj jh n σ− +=

where hj is the bandwidth for the jth dimension, j=1,..,d (d=2 in this paper), n the number of observations and jσ is the standard deviation of the observations in the jth

dimension. This is a simple method, which is based on providing the optimal bandwidth for a reference distribution, in this case a multivariate normal distribution.

3. Kernel smoothing crime over London

Crime maps were generated for each month from 03/2011-08/2011 for vehicle crime. These are displayed for London in Figure 1. The scale gives the average number of events in that area.

Figure 1: Vehicle crime in London between 03/2011 and 08/2011

This figure demonstrates the power of this technique. It can be seen that while there is some variance in crime levels between different months, the hot spots remain in similar locations. In particular, vehicle crime remains high in north east London

throughout the six month period considered.

Figure 2: Vehicle crime in Camden 03/11-08/11

Figure 2 displays the results for vehicle crime in Camden. The figure also includes the recorded crime events for that month, represented by circles, which are sized proportional to the number of recorded events at the centre of the circle. This allows a comparison between currently available crime visualisations and the new method.

It is clear that over small areas crime is more variable, which is possibly unsurprising, as smaller areas will show small changes each month, while across London these minor changes will be averaged out. In particular Figure 4 shows a large spike in vehicle crime in March (25 events occurred on one road), which is reflected in the smoothing method.

The contours help act as an informative summary of the point data, providing more information on the areas in which most events are occurring.

4. Conclusions and future research

This investigation has demonstrated that kernel smoothing methods can be a powerful tool for visualising and understanding patterns in crime data. The visualisations displayed here are limited to vehicle crime but can be applied to any crime type. This method could be applied to additional data sets, such as mortality data and business data; both data sets have had smoothing techniques applied them in the past. Potentially any data set with a geographical element could have kernel smoothing methods applied to it.

The method allows users to compare the crime both over time and within an area; this provides advantages when looking for patterns which may be less apparent in point

data. It also potentially aids in avoiding disclosure issues; if data is released only in smoothed format then it might be more difficult to identify individuals.

Kernel estimates are relatively easy to calculate in R, which is freely available. Interpreting patterns and differences between areas can be much more difficult when only point estimates are available. This method has been put to exploratory use here, but it can potentially be used in an analytic fashion, see Diggle (2003).

Additional visualisations could be used to overlay the kernel smoothing maps over actual geography, such as Google maps, allowing users to see the crime levels over laid on their areas. This will enable users to look for explanations of why an area has a low or high level of vehicle crime: a car park or busy road might have more vehicle crime associated with it, for example.

5. References

Baddeley, A. and Turner, R. (2005) “Spatstat: An R package for analyzing spatial point patterns,” Journal of Statistical Software, 12 1-42 Baddeley, A. (2010) “Analysing spatial point patterns in R,” Technical report, CSIR. Version 4 Cabinet Office (2012) “Open data white paper: unleashing the potential,” https://www.gov.uk/government/publications/open-data-white-paper-unleashing-the-potential Carrington, A., Heady, P., Ralphs, M., Rahman, N. and Mitchell, B. (2007) “Smoothing of standardised mortality ratios,” National Statistics Methodological Series No 35 Diggle, P.J.(1985) “A kernel method for smoothing point process data,” Applied Statistics (Journal of the Royal Statistical Society series C) 34 138-147 Diggle. P.J. (2003) Statistical analysis of spatial point patterns, Hodder Arnold, London, second edition Epanechnikov, V.A. (1969) “Non-parametric estimation of a multivariate probability density,” Theory of Probability and its Applications 14: 153–158 Hand, D. (2012) “Open data: problems and promise,” Methods News, Newsletter for the ESRC National Centre for Research Methods, Summer 2012 Nadaraya, E. A. (1964) “On estimating regression,” Theory of Probability and its Applications 9 (1): 141–2 Ralphs, M. (2011). “Using kernel density estimation of inter-departmental business register data to explore functional economic geographies in greater London,” Internal ONS report Sheather, S.J. (1992) “The performance of six popular bandwidth selection methods on some real data sets (with discussion),” Computational Statistics 7: 225–250, 271–281 Scott, D.W. (1992) Multivariate density estimation. theory, practice and visualization, New York: Wiley Simonoff, J.S. (1996) Smoothing methods in statistics, Springer